Exponential decay in a thermoelastic mixture of solids

International Journal of Solids and Structures 46 (2009) 1659–1666

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Exponential decay in a thermoelastic mixture of solids M.S. Alves a, J.E. Muñoz Rivera b, R. Quintanilla c,* a b c

Departamento de Matemática, Universidade Federal de Viçosa - UFV, 36570-000, Viçosa - MG, Brazil Laboratório Nacional de Computação Científica - LNCC, Av. Getúlio Vargas, 333, 25651-075, Petrópolis - RJ, Brazil Universidad Politecnica de Catalunya, Departamento de Matemática Aplicada 2, UPC C. Colón 11, 08222 Terrassa, Barcelona, Spain

a r t i c l e

i n f o

Article history: Received 18 July 2008 Received in revised form 11 November 2008 Available online 24 December 2008 Keywords: Thermoelastic mixtures Exponential decay Weakly coupled system

a b s t r a c t In this paper, we investigate the asymptotic behaviour of solutions to the initial boundary value problem for a one-dimensional mixture of thermoelastic solids. Our main result is to establish a necessary and sufficient condition over the coefficients of the system to get the exponential stability of the corresponding semigroup. We also prove the impossibility of time localization of solutions. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Thermoelastic mixtures of solids is a subject which has deserved much attention in the recent years. The first works on this topic were the contributions by Truesdell and Toupin (1960), Green and Naghdi (1965, 1968) or Bowen and Wiese (1969). Presentations of these theories can be found in the articles Atkin and Craine (1976), Bedford and Drumheller (1983) or the books by Bowen (1976), and Rajagopal and Tao (1995). The theories of mixtures of elastic solids presented in Bowen (1976), Green and Steel (1966) and Steel (1967) use as independent constitutive variables the displacement gradients and the relative velocity, and the spatial description is used. The first theory based on the Lagrangian description has been presented by Bedford and Stern (1972). There, the independent constitutive variables are the displacement gradients and the relative displacement. In recent years, an increasing interest has been directed to the study of the qualitative properties of this theory. In particular, we can recall several results concerning existence, uniqueness, continuous dependence and asymptotic stability (see Iesßan and Quintanilla, 1994; Martínez and Quintanilla, 1995). In this paper, we want to emphasize the study of the decay of solutions to the case of a one-dimensional beam composed by a mixture of two thermoelastic solids and we want to know when we can expect exponential stability for our system. Through the paper, to simplify our expressions, we speak several times about slow decay or exponential decay of the solutions. We will say that the decay * Corresponding author. Tel.: +34 93 739 8162; fax: +34 93 739 8101. E-mail addresses: [email protected] (M.S. Alves), [email protected] (J.E. Muñoz Rivera), [email protected] (R. Quintanilla). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.12.005

of the solutions is exponential if they are exponentially stable and, if they are not, we will say that the decay of the solutions is slow. Perhaps it is worth recalling the main difference between these two concepts in a thermomechanical context. If the decay is exponential, then after a short period of time, the thermomechanical displacements are very small and can be neglected. However, if the decay is slow, then the solutions weaken in a way that thermomechanical displacements could be appreciated in the system after some time. Therefore, the nature of the solutions highly determines the temporal behavior of the system and, from a thermomechanical point of view, it is relevant to be able to classify them. We consider a beam composed by a mixture of two interacting continua that occupies the interval ð0; LÞ. The displacements of typical particles at time t are u and w, where u ¼ uðx; tÞ; w ¼ wðy; tÞ; x; y 2 ð0; LÞ. We assume that the particles under consideration occupy the same position at time t ¼ 0, so that x ¼ y. The temperature in each point x and the time t is given by h ¼ hðx; tÞ. We denote by qi the mass density of each constituent at time t ¼ 0, T; S the partial stresses associated with the constituents, P the internal diffusive force, N the entropy density, Q the heat flux vector and T 0 is the absolute temperature in the reference configuration. In the absent of body forces the system of equations which governs the linear theory consists of the equations of motion (see Iesßan, 1991)

€ ¼ Sx þ P; q1 u€ ¼ T x  P; q2 w

ð1Þ

the energy equation

ðq1 þ q2 ÞT 0 N_ ¼ Q x ; and the constitutive equations

ð2Þ

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T ¼ a11 ux þ a12 wx þ b1 h;

S ¼ a12 ux þ a22 wx þ b2 h;

P ¼ aðu  wÞ; N ¼ b1 ux  b2 wx þ ch;

ð3Þ

Q ¼ Khx :

ð4Þ

If we substitute the constitutive equations into the motion equations and the energy equation, we obtain the system of field equations

q1 u€  a11 uxx  a12 wxx þ aðu  wÞ  b1 hx ¼ 0 in ð0; LÞ  ð0; 1Þ; €  a12 uxx  a22 wxx  aðu  wÞ  b2 hx ¼ 0 in ð0; LÞ  ð0; 1Þ; q2 w _ _ x ¼ 0 in ð0; LÞ  ð0; 1Þ; ch  jhxx  b1 u_ x  b2 w ð5Þ KT 1 0 ð

1

where j ¼ q1 þ q2 Þ . We assume that the constants q1 , q2 , c, j, and a are positive, and that the matrix ðaij Þ is positive definite. Thus, we assume that the coefficients of the matrix verify

a11 > 0;

a11 a22  a212 > 0:

ð6Þ

We are going to study the problem proposed by the system (5), with the following initial conditions

_ _ uð:;0Þ ¼ u0 ; uð:;0Þ ¼ u1 ; wð:;0Þ ¼ w0 ; wð:;0Þ ¼ w1 ; hð:;0Þ ¼ h0

ð7Þ

with null displacements and heat flux at the boundary. That is, we impose boundary data uð0;tÞ ¼ uðL;tÞ ¼ wð0;tÞ ¼ wðL;tÞ ¼ 0; in ð0;1Þ;hx ð0;tÞ ¼ hx ðL;tÞ ¼ 0 in ð0;1Þ: ð8Þ

Our purpose in this work is to investigate the stability and regularity of the solutions to system (5), (7), (8). The asymptotic behaviour as t ! 1 of solutions to the equations of linear thermoelasticity has been studied by many authors. We refer to the book of Liu and Zheng (1999) for a general survey on those topics. However, the case of the thermoelastic mixtures has been only studied at Martínez and Quintanilla (1995). There, the authors prove the asymptotic stability (generically). We note that we can not expect that the solutions to this system always decay in a exponentially way. For instance, in case that b1 þ b2 ¼ 0 and q2 ða11 þ a12 Þ ¼ q1 ða12 þ a22 Þ, we can obtain solutions of the form u ¼ w and h ¼ 0. These solutions are undamped and do not decay to zero. Also when b1 ¼ b2 ¼ 0, the mechanical and thermal parts are not coupled and the displacements do not decay. Moreover, we will see that if

m2 p2 L

2

¼

aððq1 b22  q2 b21 Þ þ b1 b2 ðq1  q2 ÞÞ ; b1 b2 ðq2 a11  a22 q1 Þ  a12 ðb21 q2  b22 q1 Þ

ð9Þ

holds for some m 2 N, then there exists an undamped solution. These are very particular cases, but we will see that there are some other cases where the solutions decay, but the decay is not so fast to be controlled by an exponential. From a thermomechanical point of view it happens that in these cases the heat can not control uniformly the elastic deformations of high order. Our main result is showing that the semigroup associated is exponentially stable if and only if condition (9) does not hold and

b2 ðb1 q2 a11 þ b2 q1 a12 Þ–b1 ðb2 q1 a22 þ b1 q2 a12 Þ:

view. Furthermore, we recall that for the elasticity with voids two dissipative mechanisms are needed to obtain exponential stability (Magaña and Quintanilla, 2006; Muñoz-Rivera and Quintanilla, 2008; Pamplona et al., 2009; Quintanilla, 2003), but this is not the case for the mixtures of elastic solids (as we will see). In contrast with the spatial behavior, the time behavior of solutions depends on the kind of the coupling and dissipation mechanisms. For the theories of thermoelastic solids with voids and thermoelastic mixtures of solids they are very different. It is worth noting the mathematical interest of the results proposed here. We note that our system of equation (5) is composed of three equations. Two of them are conservative and only one is dissipative. The coupling between the thermal part and the mechanical part is proposed by the parameters b1 and b2 and the two mechanical equations are also coupled between them. The question proposed in this paper is if the coupling determined by the parameters bi may bring the dissipation of the thermal effects to the mechanical part in a success way. That is, we prove the exponential stability of the solutions for the one-dimensional problem of a mixture of thermoelastic solids. We will prove that generically the coupling is so strong that the thermal dissipation brings to the system to the exponential decay. On the other hand we can ask ourselves if the decay could be faster in the sense that the solutions could vanish in a finite time. We will prove that this is not possible. That is, we show the impossibility of time localization of solutions. This paper is organized as follows: In Section 2, we establish the well posedness of the system. In Section 3, we show the exponential stability of the corresponding semigroup provided (9) does not hold and (10) holds. We show the lack of the exponential stability when relation (10) fails in Section 4. Our main tool is the theorem by Gearhart (1978) and Huang (1985) and Pruss (1984) as well as the spectral arguments. Finally we prove the impossibility of time localization of solutions.

2. Existence and uniqueness of solutions The aim of this section is to prove the existence and uniqueness of solutions of the problem determined by a thermoelastic beam. It is worth noting that a similar result was obtained in Martínez and Quintanilla (1995). But it is, suitable to state it here, because we will consider the case of functions which take values in the complex field. Let us consider the vectorial space

H ¼ H10 ð0; LÞ  H10 ð0; LÞ  L2 ð0; LÞ  L2 ð0; LÞ  L2 ð0; LÞ where

L2 ð0; LÞ ¼ fu 2 L2 ð0; LÞ;

Z

uðxÞdx ¼ 0g

0

The vectorial space H with the inner product D

~ ; w; ~ v e;g e; e ðu;w; v ; g;hÞ;ðu hÞ

E

¼

H

ð10Þ

As the theory of mixtures is one of the nonclassical elastic theories, this paper is a contribution to understand the time behavior of the solutions of nonclassical elastic theories. We recall that very few contributions have been addressed to this objective. However, in the recent years there has been developed some works for the case of elastic materials with voids (see Casas and Quintanilla, 2005a,b; Magaña and Quintanilla, 2006; Muñoz-Rivera and Quintanilla, 2008; Pamplona et al., 2009; Quintanilla, 2003). We pay attention that the kind of coupling of elasticity with voids and the mixtures of elastic materials is very different. Thus, thought both theories belong to the family of the nonclassical elastic theories, they propose different problems from the physical and mathematical points of

L

Z L  ~ x þa12 ðux w ~ x þwx u ~ x Þ þa22 wx w ~ x dx a11 ux u 0

þa

Z 0

þ q2

L

~  wÞdxþ ~ ðu wÞðu q1

Z 0

L

g ge dxþc

Z

L

Z

L

0

v ve dx

he hdx:

0

is a Hilbert space. The corresponding norm is given as

kðu; w; v ; g; hÞk2H ¼

Z

L

ða11 ux ux dx þ a12 ðux wx þ wx ux Þ þ a22 wx wx Þdx

0

þa

Z 0

þ q2

L

ðu  wÞðu  wÞdx þ q1

Z 0

L

ggdx þ c

Z

L

hhdx: 0

Z 0

L

vv dx

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

kðu; w; v ; g; hÞk2H P C 0 þ

Z

Z

L

jux j2 dx þ

0 L

0

where C 0 ¼

a a a2 minf 22 2a1122 12

L

jwx j2 dx þ

Z

0

Z

jgj2 dx þ

Z L

L

g ¼ g in

jv j2 dx

0 0

2

I 0

a q1 ð:Þxx þ q1 I a22 a q2 ð:Þxx  q2 I

2

0 I

0

0 0

C C C C q1 ð:Þx C C b2 C q2 ð:Þx A j ð:Þ xx c

0

ð:Þx

b2 c

1

b1

0

0 b1 c

0

ð:Þx

ð11Þ

ð17Þ

jhxx ¼ cq  b1 fx  b2 g x

ð18Þ

in L2 ð0; LÞ. Thus, we know that there is a unique h 2 V satisfying (18). It follows from (15) and (16) that we have to get u; w satisfying:

a11 uxx þ a12 wxx  aðu  wÞ ¼ q1 h  b1 hx |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

in L2 ð0; LÞ

ð19Þ

a12 uxx þ a22 wxx þ aðu  wÞ ¼ q2 p  b2 hx |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

in L2 ð0; LÞ

ð20Þ

~f

g~

As the sesquilinear form B : H10 ð0; LÞ  H10 ð0; LÞ ! C given by

whose domain is given by

~; wÞÞ ~ ¼ Bððu; wÞ; ðu

2

DðAÞ ¼fU ¼ ðu; w; v ; g; hÞ 2 H; u; w 2 H ð0; LÞ;

Z

L

  ~ x þ a12 ðux w ~ x þ wx u ~ x Þ þ a22 wx w ~ x dx a11 ux u

0

v ; g 2 H10 ð0; LÞ; h 2 Vg:

þa

So that, the initial - boundary value problem (5),(7),(8) can be rewritten as the following initial value problem

d UðtÞ ¼ AUðtÞ; dt

ð16Þ

Using (13) and (14) in (17) we have

; q1 ; q2 ; cg.

a12

0

in L2 ð0; LÞ

b1 v x þ b2 gx þ jhxx ¼ cq in L ð0; LÞ

0 0

B B B a11 a B A ¼ B q1 ð:Þxx  q1 I B a12 B ð:Þxx þ a I q @q

ð15Þ

2

Now, we also consider the Hilbert space V ¼ fu 2 H2 ð0; LÞ \ L2 ð0; LÞ; ux 2 H10 ð0; LÞg with norm kukV ¼ kuxx k and define the linear operator A : DðAÞ  H ! H

0

ð14Þ

a12 uxx þ a22 wxx þ aðu  wÞ þ b2 hx ¼ q2 p

0 a a a2 ; 22 2a1111 12

ð13Þ

H10 ð0; LÞ

a11 uxx þ a12 wxx  aðu  wÞ þ b1 hx ¼ q1 h in L2 ð0; LÞ

0

 jhj2 dx

in H10 ð0; LÞ

v¼f

Using Cauchy–Schwartz and Young inequality we get

Uð0Þ ¼ U 0

L

~  wÞdx ~ ðu  wÞðu

0

is continuous and coercive, it follows using the Lax–Milgram theorem (see Brezis, 1983) that given ð~f ; g~Þ in L2 ð0; LÞ  L2 ð0; LÞ there exists a unique vector function ðu; wÞ in H10 ð0; LÞ  H10 ð0; LÞ such that

Bððu; wÞ; ðu; wÞÞ ¼ 

where UðtÞ ¼ ðuðtÞ; wðtÞ; v ðtÞ; gðtÞ; hðtÞÞ0 and U 0 ¼ ðu0 ; w0 ; u1 ; w1 ; h0 Þ0 , and the prime is used to denote the transpose. From now on, we will use k:k to denote the L2 -norm and k:kH to denote the norm in the Hilbert space H. In some places we will use the expression k:kjLðH Þ to denote the norm in the space of continuous linear functions in H.

Z

Z

L

~f udx 

Z

0

L

g~wdx

0

for all ðu; wÞ in H10 ð0; LÞ  H10 ð0; LÞ. Thus a11

Z

L

ux ux dx þ a12

0

a12

Z

L

0

ux wx dx þ a22

Z

L

0 Z L 0

wx ux dx þ a wx wx dx  a

Z

Z

L

uudx  a

0 L

0

uwdx þ a

Z

Z

L

wudx ¼ 

0 L

wwdx ¼ 

0

Z

Z

L

~f udx;

0 L

g~wdx;

0

Lemma 2.1. The operator A is the infinitesimal generator of a C 0 semigroup of contractions, denoted by SA ðtÞ.

for all u; w 2 H10 ð0; LÞ and we can show that

Proof. Observe that DðAÞ ¼ H. We will show that A is a operator dissipative and that 0 belongs to the resolvent of A, then our conclusion will follow using the well known Lumer–Phillips theorem (see Pazy, 1983). We observe that if U 2 DðAÞ then

for a positive constant C. Therefore, it follows that a11 u þ a12 w and a12 u þ a22 w belong to H10 ð0; LÞ \ H2 ð0; LÞ and since

hAU;U iH ¼ a11

Z

L

v x ux dx þ a12

0

þ a12 þa

L

v x wx dx þ a

0

Z

Z

Z

L

gx ux dx þ a22

0 L

gwdx þ

0

þ b1 hx v dx þ þ b2 hx gdx þ

Z

Z Z

Z

L

v udx  a

Z

0

Z

L 0

gx wx dx  a

Z

v wdx

RehAU; U iH ¼ j

L

L

and

w¼

gudx

0

a12 a11 ða11 u þ a12 wÞ þ ða12 u þ a22 wÞ; a11 a22  a212 a11 a22  a212

we deduce that u; w 2 H10 ð0; LÞ \ H2 ð0; LÞ and satisfy (19) and (20). It is easy to show that kUkH 6 CkFkH , for a positive constant C, and we conclude that 0 belongs to the resolvent of A. h

L

½ða11 uxx  auÞ þ ða12 wxx þ awÞ

0 L

½ða12 uxx þ auÞ þ ða22 wxx  awÞ

From 2.1 we can state the following result (see Pazy, 1983).

L

Proposition 2.2. For any U 0 ¼ ðu0 ; w0 ; u1 ; w1 ; h0 Þ 2 H there exists a unique solution UðtÞ ¼ ðuðtÞ; wðtÞ; v ðtÞ; gðtÞ; hðtÞÞ of (5),(7),(8) satisfying

ðb1 v x þ b2 gx þ jhxx Þhdx:

jhx j2 dx 6 0;

a22 a12 ða11 u þ a12 wÞ  ða12 u þ a22 wÞ a11 a22  a212 a11 a22  a212

0

From there results that

Z

u¼

L

0

0

kux k þ kwx k 6 Cðk~f k þ kg~kÞ;

u; w 2 Cð½0; 1Þ; H10 ð0; LÞÞ \ C 1 ð½0; 1Þ; L2 ð0; LÞÞ; ð12Þ

0

and therefore the operator A is dissipative. Given F ¼ ðf ; g; h; p; qÞ 2 H, we must show that there exists a unique U ¼ ðu; w; v ; g; hÞ in DðAÞ such that AU ¼ F, that is:

h 2 Cð½0; 1Þ; L2 ð0; LÞÞ \ L2 ðð0; 1Þ; H1 ð0; LÞÞ: However, if U 0 2 DðAÞ hence u;w 2 Cð½0;1Þ; H10 ð0; LÞ \ H2 ð0; LÞÞ \ C 1 ð½0; 1Þ;H10 ð0; LÞÞ \ C 2 ð½0; 1Þ; L2 ð0; LÞÞ; h 2 Cð½0; 1Þ; VÞ \ C 1 ð½0; 1Þ; L2 ð0; LÞÞ:

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

3. The exponential stability Here, we will prove the exponential stability when (9) is not verified and (10) holds. Our main tool will be the following theorem, we refer to Pruss (1984) for the proof. Theorem 3.1. Let SðtÞ be a C 0 -semigroup of contractions of linear operators on a Hilbert space K with infinitesimal generator B with resolvent set qðBÞ. Then SðtÞ is exponentially stable if and only if

ðaÞ iR ¼ fib; b 2 Rg  qðBÞ; ðbÞ

jbj!þ1

Lemma 3.2. Suppose that (9) does not hold and that (10) holds. Then iR ¼ fik; k 2 Rg is contained in qðAÞ. Proof. We show this result by a contradiction argument. Suppose that there exists k 2 R such that k–0, ik is in the spectrum of A. It can be easily seen that A1 is compact, then ik must be an eigen value of A. Thus there is a vector function U ¼ ðu; w; v ; g; hÞ 2 DðAÞ, U–0, such that ikU  AU ¼ 0 or equivalently

iku  v ¼ 0

ð21Þ

ikw  g ¼ 0

ð22Þ

ikq1 v  ða11 uxx  auÞ  ða12 wxx þ awÞ  b1 hx ¼ 0

ð23Þ

ikq2 g  ða12 uxx þ auÞ  ða22 wxx  awÞ  b2 hx ¼ 0

ð24Þ

ikch  b1 v x  b2 gx  jhxx ¼ 0

ð25Þ

RehðkiI  AÞU; Ui ¼ j

Z

L

Remark 3.3. We observe that when (9) holds and

n¼

jhx j2 dx ¼ 0;

Theorem 3.4. Suppose that (9) does not hold and that (10) holds, then the semigroup SA ðtÞ is exponentially stable. Proof. We will show that

lim sup kðikI  AÞk < 1: jkj!1

Given k 2 R and F ¼ ðf ; g; h; p; qÞ 2 H, there exists U ¼ ðu; w; v ; g; hÞ 2 DðAÞ, such that ðikI  AÞU ¼ F, that is,

iku  v ¼ f in H10 ð0; LÞ

ð33Þ

H10 ð0; LÞ

ð34Þ

ikw  g ¼ g in

ikq1 v  ða11 uxx  auÞ  ða12 wxx þ awÞ  b1 hx ¼ h in L2 ð0; LÞ 2

ikq2 g  ða12 uxx þ auÞ  ða22 wxx  awÞ  b2 hx ¼ p in L ð0; LÞ

ð36Þ

ikch  b1 v x  b2 gx  jhxx ¼ q in L2 ð0; LÞ

ð37Þ

Note that

and hence h ¼ 0. From (25) we have

RehðikI  AÞU; UiH ¼ j

b1 v x þ b2 gx ¼ 0 in L2 ð0; LÞ; then b1 v þ b2 g ¼ 0 and it results from equations (21) and (22) that

b1 u þ b2 w ¼ 0:

ð27Þ

Combining of (21)–(24) yields 2

ð28Þ

2

ð29Þ

 q1 k u  ða11 uxx  auÞ  ða12 wxx þ awÞ ¼ 0;  q2 k w  ða12 uxx þ auÞ  ða22 wxx  awÞ ¼ 0:

Z

L

    b a11  b1 a12 aðb1 þ b2 Þ  2 uxx þ k2 þ u ¼ 0; q1 b2 q1 b2     b a22  b2 a12 aðb1 þ b2 Þ uxx þ k2 þ u ¼ 0:  1 q2 b1 q2 b1

L

jv j2 dx¼

ð31Þ Z

L

2

jgj dx ¼

0

ðb1 þ b2 Þðb2 a11  b1 a22  b1 a12 þ b2 a12 Þ b1 b2 ðq2 a11  a22 q1 Þ  a12 ðb21 q2  b22 q1 Þ

6 0;

ð32Þ

(30) and (31) have only the solution u ¼ 0. In this case iR  qðAÞ. If (32) is not true, then u is also solution of the equation

b22

q2  q1 ÞÞuxx

þ aðq2 b1  q1 b2 Þðb1 þ b2 Þu ¼ 0: This equation does not have non trivial solutions in general. However, we point out that if it has non trivial solutions then must there is m0 2 N for which (9) holds and we have a contradiction. Therefore iR  qðAÞ.

0

jhx j2 dx ¼ RehF; UiH

jhx j2 dx 6 C 1 kFkH kUkH ;

0

ð30Þ

We note that when

L

ð38Þ

for a positive constant C 1 . Taking the inner product in L2 ð0; LÞ of (35) with u, (36) with w and using (33) and (34) we obtain

Z

If b1 b2 –0, substituting w ¼  bb12 u in (28) and (29) we get

Z

and thus

0

a12 ðb21

ð35Þ

ð26Þ

0

 ðb1 b2 ðq2 a11  a22 q1 Þ 

aðb1 þ b2 Þðb2 a11  b1 a22  b1 a12 þ b2 a12 Þ >0 b1 b2 ðq2 a11  a22 q1 Þ  a12 ðb21 q2  b22 q1 Þ



then uðxÞ ¼ sin m0Lpx , w ¼  bb12 u and h ¼ 0 is solution of (21)–(25) pffiffiffi for k ¼ n. Moreover if b1 ¼ b2 ¼ 0, iR is not contained in the resolvent set of A, because the system (28) and (29) has infinite solutions.

lim sup kðibI  BÞ1 kLðKÞ < 1:

From (12), we have

If case that b1 vanishes, we obtain that w ¼ 0 and then a12 uxx þ au ¼ 0 which contradicts that (9) does not hold. In case b2 ¼ 0, it follows by (10) that b1 a12 –0 and our conclusion is the same. h

a11

Z

L

jux j2 dxþ

a q1

Z

L

juj2 dxþ

a12

Z

L

w u dx

q1 0 q1 0 x x 0 Z L Z L Z Z L a b 1 L wudx 1 hx udx v f dx hudx; ð39Þ  q1 0 q1 0 q1 0 0

a12

Z

L

ux wx dx 

a q2

Z

L

uwdx þ

a22

Z

L

2

jw j dx

q2 0 q2 0 x 0 Z L Z L Z L Z a b 1 L 2 þ jwj dx  2 hx wdx  ggdx  pwdx: ð40Þ q2 0 q2 0 q2 0 0

It follows by equations (33), (34) and (37) that Z

L 0

jb1 ux þ b2 wx j2 dx Z

Z L b1 uxx þ b2 wxx hðb1 ux þ b2 wx Þdx dx þ c k 0 0 Z L Z L q 1 ðb1 ux þ b2 wx Þdx þ ðb f þ b2 gÞðb1 ux þ b2 wx Þdx; þi ik 0 1 0 k

¼ ij

L

hx

and by equations (35) and (36) we obtain

ð41Þ

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

b1 uxx þ b2 wxx b1 b ¼i ðq1 a22 v  q2 a12 gÞ þ i 2 ðq1 a12 v þ q2 a11 gÞ k e e b1 b2  ða22 h  a12 pÞ  ða11 p  a12 hÞ ek ek ab b þ 1 ða22 þ a12 Þðu  wÞ  1 ðb1 a22  b2 a12 Þhx ek ek ab2 b2  ða þ a Þðu  wÞ  ða11 b2  a12 b1 Þhx ; ð42Þ ek 12 11 ek

L

¼c

jb1 v þ b2 gj2 dx ¼  c

Z

L

hb1 ux þ b2 wx dx þ i

0

Z

L

0

Z

Z

L

jhx j2 dx þ

0

1 ik

Z

q b ux þ b2 wx dx k 1

0

j c

þj

Z

0

L

hx ðb1 v þ b2 gÞdx ¼

Z

0

Z

L

nx ðb1 v þ b2 gÞdx þ

Z

0

þj

L

L

hx ðb1 v þ b2 gÞdx ¼ 0

Z 0

þ



b2

q1 q2

q1 q2

0

0

L

nx ðu  wÞdx:

0

ð47Þ

q1

q2

q1

q2

q1

q2

q1

q2

L

qðb1 u1 þ b2 /1 Þdx:

Z L 0

0

q1

q2

q1

q2

    b1 b b1 b b b hþ 2 p a11 þ 2 a12 u  1 a12 þ 2 a22 wdx:

q1

q2

q1

q2

q1

q2

ð48Þ By (33) and (34) we deduce that u  w ¼ f g þ v kig and substituting ki in (48) we have

ð43Þ

ð44Þ

jb1 v þ b2 gj2 dx

   2 Z L  b1 b2 b1 b2 u wx dx a þ a þ a þ a 11 12 x 12 22 q q q q 0 1 2 1 2 Z     b1 1 b1 b2 L b2 b b 6  jf  gj a11 þ a12 u þ 1 a12 þ 2 a22 w dx jkj q1 q2 0 q1 q2 q1 q2 Z     b 1 b b L b b b þ 1  2 jv  gj 1 a11 þ 2 a12 u þ 1 a12 þ 2 a22 w dx jkj q1 q2 0 q1 q2 q1 q2     Z L b1 b b b 2 1 2 þ jb1 v þ b2 gj a þ a v a þ a g dx q1 11 q2 12 q1 12 q2 22 0     Z L b b b b þ jb1 v þ b2 gj 1 a11 þ 2 a12 f  1 a12 þ 2 a22 g dx q1 q2 q1 q2 0 !Z     L b b21 b22 b b b þ þ jhx j 1 a11 þ 2 a12 u  1 a12 þ 2 a22 w dx q1 q2 0 q1 q2 q1 q2     Z L b1 b b b b b h þ 2 p 1 a11 þ 2 a12 u  1 a12 þ 2 a22 w dx: þ 0

q1

q2

q1

q2

q1

q2

L

qðb1 u1 þ b2 /1 Þdx:

ð45Þ

On the other hand, multiplying the equations (35) by qb1 and (36) by 1 b2 q and summing the result we obtain 2

    b b b b ikðb1 v þ b2 gÞ ¼ 1 a11 þ 2 a12 uxx þ 1 a12 þ 2 a22 wxx q1 q2 q1 q2 !   b1 b2 b21 b22 b b ðu  wÞ þ hx þ 1 h þ 2 p: þa  þ

q1 q2

b1

Z

b2

q2

qðb1 u1 þ b2 /1 Þdx !Z L b21 b22 nx pdx  þ nx hx dx 0

L

    b b b b ðb1 v þ b2 gÞ 1 a11 þ 2 a12 v þ 1 a12 þ 2 a22 gdx

q1 q2

0

Z

nx hdx  c

0

Z

q2

L

    b b b b ðb1 v þ b2 gÞ 1 a11 þ 2 a12 f þ 1 a12 þ 2 a22 gdx þ q1 q2 q1 q2 0 !Z     2 2 L b1 b2 b b b b þ þ hx 1 a11 þ 2 a12 u  1 a12 þ 2 a22 wdx

RL Considering n 2 H2 ð0; LÞ, 0 nðsÞds ¼ 0, solution of the problem nxx ¼ h, nx ð0Þ ¼ 0, nx ðLÞ ¼ 0, it follows from (44) that

 ikc

L

0

0

L

0

q1 q2

L

Z

jb1 v þ b2 gj2 dx

Z

b1

q1



¼

for k–0. Rx Rx Let us u1 ðxÞ ¼ 0 v ðsÞds and /1 ðxÞ ¼ 0 gðsÞds. Taking the inner product of (37) with b1 u1 þ b2 /1 and integrating by parts we obtain

0

hx ðb1 v þ b2 gÞdx þ Z

q1

Z

   2 Z L  b1 b2 b1 b2 u w dx a þ a þ a þ a q 11 q 12 x q1 12 q2 22 x 0 1 2  Z L     b b b b b b ðu  wÞ 1 a11 þ 2 a12 u þ 1 a12 þ 2 a22 wdx  1 2

C3 C4 khx kkUjjH þ khx kkFkH jkj jkj C6 þ C 5 kUkH kFkH þ kUkH kFkH ; jkj

L

   b b a12 u þ 1 a12 þ 2 a22 wdx

2

0

Z

b2

q2

Taking the inner product of (46) with ðqb1 a11 þ qb2 a12 Þu þ ðqb1 a12 1 2 1 þ qb2 a22 Þw, we obtain

ðb1 f þ b2 gÞðb1 ux þ b2 wx Þdx:

hðb1 u1 þ b2 /1 Þdx þ

a11 þ

L

0

L

L

Z

b1

q1

kb1 v þ b2 gk2 6 C 7 khx kkUkH þ C 8 kUkH kFkH :

kb1 ux þ b2 wx k2 6 C 2 khx kkUkH þ

Z



L

hx

0

Therefore, by (38) we can conclude that there are positive constants C 2 , C 3 , C 4 , C 5 and C 6 that

ikc

Z

RL RL Since that 0 jnx j2 dx 6 L2 0 jhj2 dx, using Poincaré and Cauchy-Schwartz inequalities and (38) we can verify that there are positive constants C 7 and C 8 such that

jb L h ðq a v  q2 a12 gÞdx  1 e 0 x 1 22 Z L jb h ðq1 a12 v þ q2 a11 gÞdx  2 e 0 x Z L jb h ða h  a12 pÞdx þi 1 ek 0 x 22 Z L jb jab1 h ða p  a12 hÞdx  i þi 2 ða þ a Þ ek 0 x 11 ek 22 12 Z L jb hx ðu  wÞdx þ i 1 ðb1 a22  b2 a12 Þ  ek 0 Z L jab2 jhx j2 dx þ  ða þ a Þ ek 12 11 0 Z L jb hx ðu  wÞdx þ i 2 ða11 b2  a12 b1 Þ  ek 0 

L

 ca

jb1 ux þ b2 wx j2 dx

0

Z

e ¼ a11 a22  a212 . By substituting (42) in (41) we get

where

Z

Then substituting (46) in (45) we get

q1 q2

q1

q2

ð46Þ

Therefore we can conclude that there exist positive constants C 9 , C 10 , C 11 , C 12 and D13 , such that

kðq2 b1 a11 þ q1 b2 a12 Þux þ ðq2 b1 a12 þ q1 b2 a22 Þwx k2 C 11 6 C 9 kb1 v þ b2 gkkUkH þ C 10 kUkH kFkH þ kUkH kFkH jkj C 12 þ kUk2H þ D13 khx kkUkH : jkj

ð49Þ

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

Since

Then the semigroup SA ðtÞ associated to operator A is not exponentially stable.

ux ¼ M 1 ðb1 ux þ b2 wx Þ þ N 1 ððq2 b1 a11 þ q1 b2 a12 Þux þ ðq2 b1 a12 þ q1 b2 a22 Þwx Þ;

Proof. We will restrict our attention to the case b1 b2 –0 and b1 þ b2 –0. First of all, note that qb1 ðab111  ab122 Þ > 0 is always positive. 1 In fact, the hypothesis implies that

wx ¼ M 2 ðb1 ux þ b2 wx Þ þ N2 ððq2 b1 a11 þ q1 b2 a12 Þux þ ðq2 b1 a12 þ q1 b2 a22 Þwx Þ;

    b1 a11 a12 b a22 a12   ¼ 2 ; q1 b1 b2 q2 b2 b1

where

q2 b1 a12 þ q1 b2 a22 ; b1 ðq2 b1 a12 þ q1 b2 a22 Þ  b2 ðq2 b1 a11 þ q1 b2 a12 Þ b2 ; N1 ¼  b1 ðq2 b1 a12 þ q1 b2 a22 Þ  b2 ðq2 b1 a11 þ q1 b2 a12 Þ q2 b1 a11 þ q1 b2 a12 ; M2 ¼  b1 ðq2 b1 a12 þ q1 b2 a22 Þ  b2 ðq2 b1 a11 þ q1 b2 a12 Þ b1 ; N2 ¼ b1 ðq2 b1 a12 þ q1 b2 a22 Þ  b2 ðq2 b1 a11 þ q1 b2 a12 Þ M1 ¼

and considering, for example, b1 > 0 and b2 < 0, we have that a11  ab122 6 0 implies ab222  ab121 P 0. It results that 0 < a11 6 bb12 a12 and b1 0 < a22 6 bb21 a12 and therefore a11 a22 6 a212 , but this is contradictory to our hypotheses over ðaij Þ. Now, we consider real numbers a and b such that ba1  bb2 –0.



For any m 2 N, we take F m ¼ ð0; 0; a sin mpL x ; b sin mpL x ; 0Þ. Let us denote by U m ¼ ðum ; wm ; v m ; gm ; hm Þ the solution of ðikI  AÞU m ¼ F m , that is hence

we deduce by (43) and (49) that there exist constants C j ; j ¼ 13; . . . ; 19, such that

kux k2 þ kwx k2 6 C 13 khx kkUkH þ C 14 kb1 v þ b2 gkkUkH C 16 þ C 15 kUkH kFkH þ kUkH kFkH jkj C 17 C 18 C 19 þ khx kkUkH þ khx kkFkH þ jkUk2H ; jkj jkj jkj

ikum  v m ¼ 0; ikwm  gm ¼ 0;

mpx ; L mpx ikq2 gm  ða12 umxx þ aum Þ  ða22 wmxx  awm Þ  b2 hmx ¼ b sin ; L ikchm  b1 v mx  b2 gmx  jhmxx ¼ 0:

ikq1 v m  ða11 umxx  aum Þ  ða12 wmxx þ awm Þ  b1 hmx ¼ a sin

ð50Þ

for k–0. From (39), (40) and (50) we verify that there exist constants C j ; j ¼ 20; . . . ; 26, that





The solutions are of the form um ¼ Am sin mpL x , wm ¼ Bm sin mpL x and m p x hm ¼ C m cosð L Þ. Therefore we get

kUk2H 6 C 20 khx kkUkH þ C 21 kb1 v þ b2 gkkUkH

 q1 Am k2 þ

C 23 C 24 þ C 22 kUkH kFkH þ kUkH kFkH þ khx kkUkH jkj jkj C 25 C 26 þ khx kkFkH þ jjUk2H : jkj jkj If jkj > C 26 , it results that

    C 26 C 24 kUk2H 6 C 20 þ khx kkUkH þ C 21 kb1 v þ b2 gkkUkH 1 jkj C 26   C 23 C 25 kUkH kFkH þ þ C 22 þ khx kkFkH ; C 26 C 26 and using the Young inequality and estimates (38) and (47) we get

kUkH 6 C 27 kFkH ;

þ b2





1

kðikI  AÞ kLðHÞ 6 C; 8k 2 R; for a positive constant C.

4. The lack of exponential stability The aim of this section is to prove that the decay of solutions can not be controlled by an exponential decay in case that (10) fails. Here we will show that there exists a sequence of real number ðkm Þ with km ! 1 and a bounded sequence ðF m Þ in H such that

m ! 1:

Theorem 4.1. Suppose that

b2 ðb1 q2 a11 þ b2 q1 a12 Þ ¼ b1 ðb2 q1 a22 þ b1 q2 a12 Þ:

mp L

C m ¼ b;

mp L

Am  ib2

ð54Þ

mp L



Am k 2 þ

mp 2

kBm þ j 1 b1

L2

2

C m ¼ 0:

ð55Þ

and equation (54) by

1 m 2 p2 a11 Am þ aAm b1 b1 L2 mp a þ Cm ¼ ; b1 L

q1

ð53Þ

L

 þ

1 b1



m 2 p2 L2

1 b2

we obtain

a12 Bm  aBm



    1 m2 p2 1 m2 p2 a A  a A a B þ a B þ 12 m m 22 m m b2 b2 b2 L2 L2 mp b þ Cm ¼ : b2 L

q2

ð56Þ

Bm k2 þ

ð57Þ

Subtracting equations (56) and (57) we find

h

kðikm I  AÞ1 F m kH ! 1;

  2 2  mp a11 Am þ aAm þ a B  a B 12 m m L2

Multiplying equation (53) by

for a positive constant C 27 . Thus, if jkj > C 26 we get

Since the function k#kðikI  AÞ1 kLðHÞ is continuous, we conclude that

L2

mp

ikcC m  ib1 k

jkj > C 26 ;

kðikI  AÞ1 FkH 6 C 27 kFk; 8F 2 H ) kðikI  AÞ1 kLðHÞ 6 C 27 :

m2 p2

C m ¼ a;  2 2   2 2  mp mp  q2 Bm k2 þ a A  a A a B þ a B þ 12 m m 22 m m L2 L2 þ b1

ð51Þ



ð52Þ

 

  q m2 p2 a11 a12 q m2 p2 a12 a22  1 k2 þ 2   Am þ 2 k 2 þ 2 Bm b1 b1 b b2 b1 b2 L L    2  a a a a a b þ þ þ Am  Bm ¼  : b1 b2 b1 b2 b1 b2 We can rewrite the last equation as

  m2 p2 b1 a11 a12 q1 k2 þ 2  Am b2 b1 L q1 b1

    2 2 m p b2 a22 a12 q2 a a  k2 þ 2  Bm  þ Bm b1 b2 b1 b2 L q2 b2   a a a b Am ¼  : þ þ b1 b2 b1 b2

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

k2 þ

a11

b1

Since, by hypothesis,

q1 ð b1



m2 p2 b1 a11 a12  b2 L2 q1 b1

 ab122 Þ ¼ qb2 ðab222  ab121 Þ it follows that

Returning to equation (58), if a12 ¼ 0, we obtain

2



q1

Am 

b1  a a a b þ þ ðAm  Bm Þ ¼  : b1 b2 b1 b2

q2 b2



Cm ¼

Bm



m2

2

Taking k2 ¼ Lp2



b1



q1

and from (60) it follows that

lim Bm ¼ 



a11 b1

m!1

 ab122 we obtain

 a a a b b a  b1 b þ () Am  Bm ¼ 2 : ðAm  Bm Þ ¼  b1 b2 b1 b2 aðb2 þ b1 Þ

To simplify the notation we take

1b s ¼ abðb2 ab and r ¼pL 2 þb1 Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 a11 a12 q ð b1  b2 Þ. 1

  m2 p2 m2 p2 mp q1 k2 þ 2 a11 þ 2 a12 Bm þ b Cm L 1 L L   m 2 p2 ¼ a  q1 k2 þ 2 a11 þ a s: L

2

L b2

mp L

! b1 p2 a12 m2 þ L2 b2 a

b1 C m ¼ a 

2

L b2

s:

ð58Þ

If a12 –0, we have the equation

b1 b2 L b L2 a  b1 a12 p2 sm2  b2 L2 as Cm ¼ 2 : ðb1 þ b2 Þa12 pm ðb1 þ b2 Þp2 a12 m2

Bm þ

ð59Þ

On the other hand, by (55) it results that

b1 k

mp L

Am þ b2 k

mp L



Bm  kc  ij

p2 m2 L2

Remark 4.2. In case of b1 ¼ b2 ¼ 0 the system is conservative and therefore there is not decay. In case b1 or b2 ¼ 0, it follows from (52) that a12 ¼ 0. We can use the same procedure as in Theorem 4.1 to get (61) and (62) to have that the resolvent is not uniformly bounded. Therefore there are solutions without energy dissipation. When b1 þ b2 ¼ 0, condition (52) implies that q2 ða11 þ a12 Þ ¼ q1 ða12 þ a22 Þ. In this case there always exist undamped solutions.

ðL2 ck  ijm2 p2 Þ b1 s Cm ¼  : Lðb1 þ b2 Þpmk ðb1 þ b2 Þ

The aim of this section is to prove the impossibility of time localization of solutions of the system of the thermoelastic mixtures with the boundary conditions (8). To prove the impossibility of solutions we will show the uniqueness of solutions of the backward in time problem. Thus, it will be suitable to recall that the system of equations which govern the backward in time problem is:

q1 u€  a11 uxx  a12 wxx þ aðu  wÞ  b1 hx ¼ 0 in ð0; 1Þ  ð0; LÞ; €  a12 uxx  a22 wxx  aðu  wÞ  b2 hx ¼ 0 in ð0; 1Þ  ð0;LÞ; q2 w _ x ¼ 0 in ð0; 1Þ  ð0;LÞ: ch_ þ jhxx  b1 u_ x  b2 w

 C m ¼ 0;

and since Am ¼ Bm þ s we have

Bm 

h

5. Impossibility of localization

Substituting k in the last equation we get

Bm þ

b1 s : ðb1 þ b2 Þ

In the same way, we obtain (63).

Thus, Am ¼ Bm þ s and substituting in the equation (53) we obtain

ðb1 þ b2 Þp2 a12 m2

ða  asÞL ; b1 pm

ð60Þ

We will study the problem determined by system (64), with boundary conditions (8) and with the null initial conditions

_ uð0; :Þ ¼ 0; wð0; :Þ ¼ 0;

uð0; :Þ ¼ 0;

Subtracting (59) and (60) we get

_ wð0; :Þ ¼ 0;

hð0; :Þ ¼ 0: ð65Þ

3

Cm ¼

b2 L ða  asÞk b1 b2 L2 pmk þ ðcL2 mk  ijm3 p2 Þpa12

: Lemma 5.1. Let us assume that q1 , q2 , c, j, and a are positive, and that the matrix ðaij Þ is positive definite. Let ðu; w; hÞ be a solution of the problem determined by the system (64), the initial conditions (65) and the boundary conditions (8). Then u ¼ w ¼ h ¼ 0.

Since k ¼ rm we obtain

Cm ¼

b2 L3 ða  asÞr b1 b2 L2 prm þ ðL2 rmc  ijm2 p2 Þpa12

;

Proof. The energy equation gives

and it follows that

lim C m ¼ 0:

E1 ðtÞ ¼

m!1

b1 s lim Bm ¼  ; m!1 ðb1 þ b2 Þ

Z

L

 a11 j ux j2 þ 2a12 ux wx þ a22 j wx j2 þ aðu  wÞ2

0

ð61Þ

t 0

Z 0

L

jh2x dx ds:

If we multiply the two first equations of (64) by ut and wt , respectively, and the third one by h, we obtain

and

lim Am ¼ lim ðBm þ sÞ ¼  m!1

b1 s b2 s þs¼ : ðb1 þ b2 Þ ðb1 þ b2 Þ

ð62Þ

Finally, we know that

  kU m k2H P C 0 kumx k2 þ kwmx k2 þ kum k2 þ kgm k2 þ khm k2

Z

 a11 j ux j2 þ 2a12 ux wx þ a22 j wx j2 0  _ 2 þ q2 j wj _ 2  ch2 dx þaðu  wÞ2 þ q1 j uj Z t Z L _ x þ 2b2 uh _ x  jh2x Þdxds: ¼ ð2b1 uh

E2 ðtÞ ¼

1 2

L

0

and therefore

kU m k2H P C 0 kumx k2 ¼

1 2

Z  _ 2 þ q2 j wj _ 2 þ ch2 dx ¼ þq1 j uj

Moreover from (60) we obtain

m!1

ð64Þ

C 0 p2 ðAm mÞ2 ! 1; 2L

m ! 1:

ð63Þ

0

We need a third equality. To obtain it, we use the Lagrange identity method. For a fixed t 2 ð0; TÞ, we use the identities

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M.S. Alves et al. / International Journal of Solids and Structures 46 (2009) 1659–1666

@ _ uð2t _ € ðsÞuð2t _ _ u € ð2t  sÞ; ðq uðsÞ  sÞÞ ¼ q1 u  sÞ  q1 uðsÞ @s 1 @ _ wð2t _ € wð2t _ _ wð2t €  sÞÞ ¼ q2 wðsÞ  sÞ  q2 wðsÞ  sÞ; ðq wðsÞ @s 2 @ _ _ ðchðsÞhð2t  sÞÞ ¼ chðsÞhð2t  sÞ  chðsÞhð2t  sÞ: @s

for every t P 0 and then in view of the initial conditions, it follows that the only solution to our problem is the null solution. h Thus, we can state the following:

_ Here, uð:Þ means the derivative of u with respect to the parameter inside the parenthesis. From the field equations, the boundary conditions and the null initial conditions, we obtain the following equality

Z

L



0

¼

2

a11 j ux j þ 2a12 ux wx þ a22

Z

L

0

 j wx j þ aðu  wÞ þ ch dx 2

2

2

_ 2 þ q2 j wj _ 2 Þdx: ðq1 j uj

Z

  a11 j ux j2 þ 2a12 ux wx þ a22 j wx j2 þ aðu  wÞ2 dx:

L

References

0

Let  be a small, but positive constant. Let us consider EðtÞ ¼ E2 ðtÞ þ E1 ðtÞ. We note that

EðtÞ ¼

Z

1 2

L

0

_ 2 þ q2 j wj _ 2 þ ch2 Þ ððq1 j uj

þ ð2 þ Þða11 j ux j2 þ 2a12 ux wx þ a22 j wx j2 þ aðu  wÞ2 ÞÞdx; ð66Þ is a positive function and it defines a measure on the solution. We take  strictly less than one, but greater than zero. As

Z

EðtÞ ¼ ð1  Þ

t 0

Z

L

j j hx j2 dxds þ 2

Z

0

t

0

Z

L

_ x þ b2 wh _ x Þdxds ðb1 uh

0

we have:

dE ¼ ð1  Þ dt

Z

L

j j hx j2 dxds þ 2

0

Z

L

_ x þ b2 wh _ x Þdx ðb1 uh

ð67Þ

0

The A–G inequality implies that

Z

L

_ x þ b2 wh _ x Þdx 6 K 1 ðb1 uh

0

Z 0

þ 1

L

_ 2 þ q2 j wj _ 2 Þdx ðq1 j uj

Z

L

j j hx j2 dx

0

where 1 is as small as we want, but positive and K 1 can be calculated in terms of the constitutive coefficients and 1 . If we take 1 6 1  , there exists a positive constant C such that

dE 6C dt

Z

L



0



_ 2 þ q2 j wj _ 2 þ ch2 dx: q1 j uj

ð68Þ

We obtain that the estimate

dE 6 CEðtÞ; dt

Acknowledgment The work of R. Quintanilla is supported by the project ‘‘Qualitative study of thermomechanical problems” (MTM2006-03706). J. Muñoz Rivera is supported by CNPq-Brazil grant 309166/2007-1. The authors thank to the anonymous referees their useful comments.

Thus, we have

E2 ðtÞ ¼

Theorem 5.2. Let ðu; w; hÞ be a solution of the problem determined by the system (5), the initial conditions (7) and the boundary conditions (8) such that u ¼ w ¼ h  0 after a finite time t0 > 0. Then u ¼ w ¼ h  0 for every t P 0.

ð69Þ

is satisfied for every t P 0. This inequality implies that EðtÞ 6 Eð0Þ expðCtÞ for every t greater than zero. As we assume null initial conditions we see that EðtÞ  0 for every t P 0. If we take into account the definition of EðtÞ, it follows that u  0, h  0 and w  0

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